How the IRS can use the Riemann Zeta function to detect tax fraud

Many systems exhibit a digit bias. For example, the first digit (base 10) of the Fibonacci numbers or 2ⁿ equals 1 about 30% of the time. This phenomena was first noticed by observing which pages of log tables were most worn with age -- it's a good thing there were no calculators 100 years ago! We show that the first digit of values of L-functions near the critical line also exhibit this bias. A similar bias exists (in a certain sense) for the first digit of terms in the 3x+1 problem, provided the base is not a power of two. For L-functions the main tool is the Log-Normal law; for 3x+1 it is the rate of equidistribution of n log_B2 mod 1 and understanding the irrationality measure of log_B2. This work is joint with Alex Kontorovich, and led to the author being contacted by the Criminal Investigative Division of the Internal Revenue Service!